Modeling of interaction of hydraulic fractures in complex fracture networks

ABSTRACT

Methods of performing a fracture operation at a wellsite with a fracture network are provided. The methods involve obtaining wellsite data and a mechanical earth model, and generating a hydraulic fracture growth pattern for the fracture network over time. The generating involves extending hydraulic fractures from a wellbore and into the fracture network of a subterranean formation to form a hydraulic fracture network, determining hydraulic fracture parameters after the extending, determining transport parameters for proppant passing through the hydraulic fracture network, and determining fracture dimensions of the hydraulic fractures from the hydraulic fracture parameters, the transport parameters and the mechanical earth model. The methods also involve performing stress shadowing on the hydraulic fractures to determine stress interference between fractures and repeating the generating based on the determined stress interference. The methods may also involve determining crossing behavior.

BACKGROUND

The present disclosure relates generally to methods and systems forperforming wellsite operations. More particularly, this disclosure isdirected to methods and systems for performing fracture operations, suchas investigating subterranean formations and characterizing hydraulicfracture networks in a subterranean formation.

In order to facilitate the recovery of hydrocarbons from oil and gaswells, the subterranean formations surrounding such wells can behydraulically fractured. Hydraulic fracturing may be used to createcracks in subsurface formations to allow oil or gas to move toward thewell. A formation is fractured by introducing a specially engineeredfluid (referred to as “fracturing fluid” or “fracturing slurry” herein)at high pressure and high flow rates into the formation through one ormore wellbores. Hydraulic fractures may extend away from the wellborehundreds of feet in two opposing directions according to the naturalstresses within the formation. Under certain circumstances, they mayform a complex fracture network. Complex fracture networks can includeinduced hydraulic fractures and natural fractures, which may or may notintersect, along multiple azimuths, in multiple planes and directions,and in multiple regions.

Current hydraulic fracture monitoring methods and systems may map wherethe fractures occur and the extent of the fractures. Some methods andsystems of microseismic monitoring may process seismic event locationsby mapping seismic arrival times and polarization information intothree-dimensional space through the use of modeled travel times and/orray paths. These methods and systems can be used to infer hydraulicfracture propagation over time.

Patterns of hydraulic fractures created by the fracturing stimulationmay be complex and may form a fracture network as indicated by adistribution of associated microseismic events. Complex hydraulicfracture networks have been developed to represent the created hydraulicfractures. Examples of fracture models are provided in U.S. Pat. Nos.6,101,447, 7,363,162, 7,788,074, U.S. Ser. Nos. 20080133186,20100138196, and 20100250215.

SUMMARY

In at least one aspect, the present disclosure relates to methods ofperforming a fracture operation at a wellsite. The wellsite ispositioned about a subterranean formation having a wellbore therethroughand a fracture network therein. The fracture network has naturalfractures therein. The wellsite may be stimulated by injection of aninjection fluid with proppant into the fracture network. The methodinvolves obtaining wellsite data comprising natural fracture parametersof the natural fractures and obtaining a mechanical earth model of thesubterranean formation and generating a hydraulic fracture growthpattern for the fracture network over time. The generating involvesextending hydraulic fractures from the wellbore and into the fracturenetwork of the subterranean formation to form a hydraulic fracturenetwork including the natural fractures and the hydraulic fractures,determining hydraulic fracture parameters of the hydraulic fracturesafter the extending, determining transport parameters for the proppantpassing through the hydraulic fracture network, and determining fracturedimensions of the hydraulic fractures from the determined hydraulicfracture parameters, the determined transport parameters and themechanical earth model. The method also involves performing stressshadowing on the hydraulic fractures to determine stress interferencebetween the hydraulic fractures and repeating the generating based onthe determined stress interference.

If the hydraulic fracture encounters a natural fracture, the method mayalso involve determining the crossing behavior between the hydraulicfractures and an encountered fracture based on the determined stressinterference, and the repeating may involve repeating the generatingbased on the determined stress interference and the crossing behavior.The method may also involve stimulating the wellsite by injection of aninjection fluid with proppant into the fracture network.

The method may also involve, if the hydraulic fracture encounters anatural fracture, determining the crossing behavior at the encounterednatural fracture, and wherein the repeating comprises repeating thegenerating based on the determined stress interference and the crossingbehavior. The fracture growth pattern may be altered or unaltered by thecrossing behavior. A fracture pressure of the hydraulic fracture networkmay be greater than a stress acting on the encountered fracture, and thefracture growth pattern may propagate along the encountered fracture.The fracture growth pattern may continue to propagate along theencountered fracture until an end of the natural fracture is reached.The fracture growth pattern may change direction at the end of thenatural fracture, and the fracture growth pattern may extend in adirection normal to a minimum stress at the end of the natural fracture.The fracture growth pattern may propagate normal to a local principalstress according to the stress shadowing.

The stress shadowing may involve performing displacement discontinuityfor each of the hydraulic fractures. The stress shadowing may involveperforming stress shadowing about multiple wellbores of a wellsite andrepeating the generating using the stress shadowing performed on themultiple wellbores. The stress shadowing may involve performing stressshadowing at multiple stimulation stages in the wellbore.

The method may also involve validating the fracture growth pattern. Thevalidating may involve comparing the fracture growth pattern with atleast one simulation of stimulation of the fracture network.

The extending may involve extending the hydraulic fractures along afracture growth pattern based on the natural fracture parameters and aminimum stress and a maximum stress on the subterranean formation. Thedetermining fracture dimensions may include one of evaluating seismicmeasurements, ant tracking, sonic measurements, geological measurementsand combinations thereof. The wellsite data may include at least one ofgeological, petrophysical, geomechanical, log measurements, completion,historical and combinations thereof. The natural fracture parameters maybe generated by one of observing borehole imaging logs, estimatingfracture dimensions from wellbore measurements, obtaining microseismicimages, and combinations thereof.

BRIEF DESCRIPTION OF THE DRAWINGS

Embodiments of the system and method for characterizing wellborestresses are described with reference to the following figures. The samenumbers are used throughout the figures to reference like features andcomponents.

FIG. 1.1 is a schematic illustration of a hydraulic fracturing sitedepicting a fracture operation;

FIG. 1.2 is a schematic illustration of a hydraulic fracture site withmicroseismic events depicted thereon;

FIG. 2 is a schematic illustration of a 2D fracture;

FIG. 3 is a schematic illustration of a stress shadow effect;

FIG. 4 is a schematic illustration comparing 2D DDM and Flac3D for twoparallel straight fractures;

FIGS. 5.1-5.3 are graphs illustrating 2D DDM and Flac3D of extendedfractures for stresses in various positions;

FIGS. 6.1-6.2 are graphs depicting propagation paths for two initiallyparallel fractures in isotropic and anisotropic stress fields,respectively;

FIGS. 7.1-7.2 are graphs depicting propagation paths for two initiallyoffset fractures in isotropic and anisotropic stress fields,respectively;

FIG. 8 is a schematic illustration of transverse parallel fracturesalong a horizontal well;

FIG. 9 is a graph depicting lengths for five parallel fractures;

FIG. 10 is a schematic diagram depicting UFM fracture geometry and widthfor the parallel fractures of FIG. 9;

FIGS. 11.1-11.2 are schematic diagrams depicting fracture geometry for ahigh perforation friction case and a large fracture spacing case,respectively;

FIG. 12 is a graph depicting microseismic mapping;

FIGS. 13.1-13.4 are schematic diagrams illustrating a simulated fracturenetwork compared to the microseismic measurements for stages 1-4,respectively;

FIGS. 14.1-14.4 are schematic diagrams depicting a distributed fracturenetwork at various stages;

FIG. 15 is a flow chart depicting a method of performing a fractureoperation; and

FIGS. 16.1-16.4 are schematic illustrations depicting fracture growthabout a wellbore during a fracture operation.

DETAILED DESCRIPTION

The description that follows includes exemplary apparatuses, methods,techniques, and instruction sequences that embody techniques of theinventive subject matter. However, it is understood that the describedembodiments may be practiced without these specific details.

Models have been developed to understand subsurface fracture networks.The models may consider various factors and/or data, but may not beconstrained by accounting for either the amount of pumped fluid ormechanical interactions between fractures and injected fluid and amongthe fractures. Constrained models may be provided to give a fundamentalunderstanding of involved mechanisms, but may be complex in mathematicaldescription and/or require computer processing resources and time inorder to provide accurate simulations of hydraulic fracture propagation.A constrained model may be configured to perform simulations to considerfactors, such as interaction between fractures, over time and underdesired conditions.

An unconventional fracture model (UFM) (or complex model) may be used tosimulate complex fracture network propagation in a formation withpre-existing natural fractures. Multiple fracture branches can propagatesimultaneously and intersect/cross each other. Each open fracture mayexert additional stresses on the surrounding rock and adjacentfractures, which may be referred to as “stress shadow” effect. Thestress shadow can cause a restriction of fracture parameters (e.g.,width), which may lead to, for example, a greater risk of proppantscreenout. The stress shadow can also alter the fracture propagationpath and affect fracture network patterns. The stress shadow may affectthe modeling of the fracture interaction in a complex fracture model.

A method for computing the stress shadow in a complex hydraulic fracturenetwork is presented. The method may be performed based on an enhanced2D Displacement Discontinuity Method (2D DDM) with correction for finitefracture height or 3D Displacement Discontinuity Method (3D DDM). Thecomputed stress field from 2D DDM may be compared to 3D numericalsimulation (3D DDM or flac3D) to determine an approximation for the 3Dfracture problem. This stress shadow calculation may be incorporated inthe UFM. The results for simple cases of two fractures shows thefractures can either attract or expel each other depending, for example,on their initial relative positions, and may be compared with anindependent 2D non-planar hydraulic fracture model.

Additional examples of both planar and complex fractures propagatingfrom multiple perforation clusters are presented, showing that fractureinteraction may control the fracture dimension and propagation pattern.In a formation with small stress anisotropy, fracture interaction canlead to dramatic divergence of the fractures as they may tend to repeleach other. However, even when stress anisotropy is large and fractureturning due to fracture interaction is limited, stress shadowing mayhave a strong effect on fracture width, which may affect the injectionrate distribution into multiple perforation clusters, and hence overallfracture network geometry and proppant placement.

FIGS. 1.1 and 1.2 depict fracture propagation about a wellsite 100. Thewellsite has a wellbore 104 extending from a wellhead 108 at a surfacelocation and through a subterranean formation 102 therebelow. A fracturenetwork 106 extends about the wellbore 104. A pump system 129 ispositioned about the wellhead 108 for passing fluid through tubing 142.

The pump system 129 is depicted as being operated by a field operator127 for recording maintenance and operational data and/or performing theoperation in accordance with a prescribed pumping schedule. The pumpingsystem 129 pumps fluid from the surface to the wellbore 104 during thefracture operation.

The pump system 129 may include a water source, such as a plurality ofwater tanks 131, which feed water to a gel hydration unit 133. The gelhydration unit 133 combines water from the tanks 131 with a gellingagent to form a gel. The gel is then sent to a blender 135 where it ismixed with a proppant from a proppant transport 137 to form a fracturingfluid. The gelling agent may be used to increase the viscosity of thefracturing fluid, and to allow the proppant to be suspended in thefracturing fluid. It may also act as a friction reducing agent to allowhigher pump rates with less frictional pressure.

The fracturing fluid is then pumped from the blender 135 to thetreatment trucks 120 with plunger pumps as shown by solid lines 143.Each treatment truck 120 receives the fracturing fluid at a low pressureand discharges it to a common manifold 139 (sometimes called a missiletrailer or missile) at a high pressure as shown by dashed lines 141. Themissile 139 then directs the fracturing fluid from the treatment trucks120 to the wellbore 104 as shown by solid line 115. One or moretreatment trucks 120 may be used to supply fracturing fluid at a desiredrate.

Each treatment truck 120 may be normally operated at any rate, such aswell under its maximum operating capacity. Operating the treatmenttrucks 120 under their operating capacity may allow for one to fail andthe remaining to be run at a higher speed in order to make up for theabsence of the failed pump. A computerized control system 149 may beemployed to direct the entire pump system 129 during the fracturingoperation.

Various fluids, such as conventional stimulation fluids with proppants,may be used to create fractures. Other fluids, such as viscous gels,“slick water” (which may have a friction reducer (polymer) and water)may also be used to hydraulically fracture shale gas wells. Such “slickwater” may be in the form of a thin fluid (e.g., nearly the sameviscosity as water) and may be used to create more complex fractures,such as multiple micro-seismic fractures detectable by monitoring.

As also shown in FIGS. 1.1 and 1.2, the fracture network includesfractures located at various positions around the wellbore 104. Thevarious fractures may be natural fractures 144 present before injectionof the fluids, or hydraulic fractures 146 generated about the formation102 during injection. FIG. 1.2 shows a depiction of the fracture network106 based on microseismic events 148 gathered using conventional means.

Multi-stage stimulation may be the norm for unconventional reservoirdevelopment. However, an obstacle to optimizing completions in shalereservoirs may involve a lack of hydraulic fracture models that canproperly simulate complex fracture propagation often observed in theseformations. A complex fracture network model (or UFM), has beendeveloped (see, e.g., Weng, X., Kresse, O., Wu, R., and Gu, H., Modelingof Hydraulic Fracture Propagation in a Naturally Fractured Formation.Paper SPE 140253 presented at the SPE Hydraulic Fracturing Conferenceand Exhibition, Woodlands, Tex., USA, Jan. 24-26 (2011) (hereafter “Weng2011”); Kresse, O., Cohen, C., Weng, X., Wu, R., and Gu, H. 2011(hereafter “Kresse 2011”). Numerical Modeling of Hydraulic Fracturing inNaturally Fractured Formations. 45th US Rock Mechanics/GeomechanicsSymposium, San Francisco, Calif., June 26-29, the entire contents ofwhich are hereby incorporated herein).

Existing models may be used to simulate fracture propagation, rockdeformation, and fluid flow in the complex fracture network createdduring a treatment. The model may also be used to solve the fullycoupled problem of fluid flow in the fracture network and the elasticdeformation of the fractures, which may have similar assumptions andgoverning equations as conventional pseudo-3D fracture models. Transportequations may be solved for each component of the fluids and proppantspumped.

Conventional planar fracture models may model various aspects of thefracture network. The provided UFM may also involve the ability tosimulate the interaction of hydraulic fractures with pre-existingnatural fractures, i.e. determine whether a hydraulic fracturepropagates through or is arrested by a natural fracture when theyintersect and subsequently propagates along the natural fracture. Thebranching of the hydraulic fracture at the intersection with the naturalfracture may give rise to the development of a complex fracture network.

A crossing model may be extended from Renshaw and Pollard (see, e.g.,Renshaw, C. E. and Pollard, D. D. 1995, An Experimentally VerifiedCriterion for Propagation across Unbounded Frictional Interfaces inBrittle, Linear Elastic Materials. Int. J. Rock Mech. Min. Sci. &Geomech. Abstr., 32: 237-249 (1995) the entire contents of which ishereby incorporated herein) interface crossing criterion, to apply toany intersection angle, and may be developed (see, e.g., Gu, H. andWeng, X. Criterion for Fractures Crossing Frictional Interfaces atNon-orthogonal Angles. 44th US Rock symposium, Salt Lake City, Utah,Jun. 27-30, 2010 (hereafter “Gu and Weng 2010”), the entire contents ofwhich are hereby incorporated by reference herein) and validated againstexperimental data (see, e.g., Gu, H., Weng, X., Lund, J., Mack, M.,Ganguly, U. and Suarez-Rivera R. 2011. Hydraulic Fracture CrossingNatural Fracture at Non-Orthogonal Angles, A Criterion, Its Validationand Applications. Paper SPE 139984 presented at the SPE HydraulicFracturing Conference and Exhibition, Woodlands, Tex., Jan. 24-26 (2011)(hereafter “Gu et al. 2011”), the entire contents of which are herebyincorporated by reference herein), and integrated in the UFM.

To properly simulate the propagation of multiple or complex fractures,the fracture model may take into account an interaction among adjacenthydraulic fracture branches, often referred to as the “stress shadow”effect. When a single planar hydraulic fracture is opened under a finitefluid net pressure, it may exert a stress field on the surrounding rockthat is proportional to the net pressure.

In the limiting case of an infinitely long vertical fracture of aconstant finite height, an analytical expression of the stress fieldexerted by the open fracture may be provided. See, e.g., Warpinski, N.F. and Teufel, L. W, Influence of Geologic Discontinuities on HydraulicFracture Propagation, JPT, Feb., 209-220 (1987) (hereafter “Warpinskiand Teufel”) and Warpinski, N. R., and Branagan, P. T., Altered-StressFracturing. SPE JPT, September, 1989, 990-997 (1989), the entirecontents of which are hereby incorporated by reference herein. The netpressure (or more precisely, the pressure that produces the givenfracture opening) may exert a compressive stress in the direction normalto the fracture on top of the minimum in-situ stress, which may equalthe net pressure at the fracture face, but quickly falls off with thedistance from the fracture.

At a distance beyond one fracture height, the induced stress may be onlya small fraction of the net pressure. Thus, the term “stress shadow” maybe used to describe this increase of stress in the region surroundingthe fracture. If a second hydraulic fracture is created parallel to anexisting open fracture, and if it falls within the “stress shadow” (i.e.the distance to the existing fracture is less than the fracture height),the second fracture may, in effect, see a closure stress greater thanthe original in-situ stress. As a result, a higher pressure may beneeded to propagate the fracture, and/or the fracture may have anarrower width, as compared to the corresponding single fracture.

One application of the stress shadow study may involve the design andoptimization of the fracture spacing between multiple fracturespropagating simultaneously from a horizontal wellbore. In ultra lowpermeability shale formations, fractures may be closely spaced foreffective reservoir drainage. However, the stress shadow effect mayprevent a fracture propagating in close vicinity of other fractures(see, e.g., Fisher, M. K, J. R. Heinze, C. D. Harris, B. M. Davidson, C.A. Wright, and K P. Dunn, Optimizing horizontal completion techniques inthe Barnett Shale using microseismic fracture mapping. SPE 90051presented at the SPE Annual Technical Conference and Exhibition,Houston, 26-29 Sep. 2004, the entire contents of which are herebyincorporated by reference herein in its entirety).

The interference between parallel fractures has been studied in the past(see, e.g., Warpinski and Teufel; Britt, L. K. and Smith, M. B.,Horizontal Well Completion, Stimulation Optimization, and RiskMitigation. Paper SPE 125526 presented at the 2009 SPE Eastern RegionalMeeting, Charleston, Sep. 23-25, 2009; Cheng, Y. 2009. Boundary ElementAnalysis of the Stress Distribution around Multiple Fractures:Implications for the Spacing of Perforation Clusters of HydraulicallyFractured Horizontal Wells. Paper SPE 125769 presented at the 2009 SPEEastern Regional Meeting, Charleston, Sep. 23-25, 2009; Meyer, B. R. andBazan, L. W, A Discrete Fracture Network Model for Hydraulically InducedFractures: Theory, Parametric and Case Studies. Paper SPE 140514presented at the SPE Hydraulic Fracturing Conference and Exhibition,Woodlands, Tex., USA, Jan. 24-26, 2011; Roussel, N. P. and Sharma, M. M,Optimizing Fracture Spacing and Sequencing in Horizontal-WellFracturing, SPEPE, May, 2011, pp. 173-184, the entire contents of whichare hereby incorporated by reference herein). The studies may involveparallel fractures under static conditions.

An effect of stress shadow may be that the fractures in the middleregion of multiple parallel fractures may have smaller width because ofthe increased compressive stresses from neighboring fractures (see,e.g., Germanovich, L. N., and Astakhov D., Fracture Closure in Extensionand Mechanical Interaction of Parallel Joints. J. Geophys. Res., 109,B02208, doi: 10.1029/2002 JB002131 (2004); Olson, J. E., Multi-FracturePropagation Modeling: Applications to Hydraulic Fracturing in Shales andTight Sands. 42nd US Rock Mechanics Symposium and 2nd US-Canada RockMechanics Symposium, San Francisco, Calif., Jun. 29-Jul. 2, 2008, theentire contents of which are hereby incorporated by reference herein).When multiple fractures are propagating simultaneously, the flow ratedistribution into the fractures may be a dynamic process and may beaffected by the net pressure of the fractures. The net pressure may bestrongly dependent on fracture width, and hence, the stress shadoweffect on flow rate distribution and fracture dimensions warrantsfurther study.

The dynamics of simultaneously propagating multiple fractures may alsodepend on the relative positions of the initial fractures. If thefractures are parallel, e.g. in the case of multiple fractures that areorthogonal to a horizontal wellbore, the fractures may repel each other,resulting in the fractures curving outward. However, if the multiplefractures are arranged in an en echlon pattern, e.g. for fracturesinitiated from a horizontal wellbore that is not orthogonal to thefracture plane, the interaction between the adjacent fractures may besuch that their tips attract each other and even connect (see, e.g.,Olson, J. E. Fracture Mechanics Analysis of Joints and Veins. PhDdissertation, Stanford University, San Francisco, Calif. (1990); Yew, C.H., Mear, M. E., Chang, C. C., and Zhang, X. C. On Perforating andFracturing of Deviated Cased Wellbores. Paper SPE 26514 presented at SPE68th Annual Technical Conference and Exhibition, Houston, Tex., Oct. 3-6(1993); Weng, X., Fracture Initiation and Propagation from DeviatedWellbores. Paper SPE 26597 presented at SPE 68th Annual TechnicalConference and Exhibition, Houston, Tex., Oct. 3-6 (1993), the entirecontents of which are hereby incorporated by reference herein).

When a hydraulic fracture intersects a secondary fracture oriented in adifferent direction, it may exert an additional closure stress on thesecondary fracture that is proportional to the net pressure. This stressmay be derived and be taken into account in the fissure opening pressurecalculation in the analysis of pressure-dependent leakoff in fissuredformation (see, e.g., Nolte, K., Fracturing Pressure Analysis fornonideal behavior. JPT, February 1991, 210-218 (SPE 20704) (1991)(hereafter “Nolte 1991”), the entire contents of which are herebyincorporated by reference herein).

For more complex fractures, a combination of various fractureinteractions as discussed above may be present. To properly account forthese interactions and remain computationally efficient so it can beincorporated in the complex fracture network model, a proper modelingframework may be constructed. A method based on an enhanced 2DDisplacement Discontinuity Method (2D DDM) may be used for computing theinduced stresses on a given fracture and in the rock from the rest ofthe complex fracture network (see, e.g., Olson, J. E., PredictingFracture Swarms—The Influence of Sub critical Crack Growth and theCrack-Tip Process Zone on Joints Spacing in Rock. In The Initiation,Propagation and Arrest of Joints and Other Fractures, ed. J. W. Cosgroveand T. Engelder, Geological Soc. Special Publications, London, 231,73-87 (2004) (hereafter “Olson 2004”), the entire contents of which arehereby incorporated by reference herein). Fracture turning may also bemodeled based on the altered local stress direction ahead of thepropagating fracture tip due to the stress shadow effect. The simulationresults from the UFM model that incorporates the fracture interactionmodeling are presented.

UFM Model Description

To simulate the propagation of a complex fracture network that consistsof many intersecting fractures, equations governing the underlyingphysics of the fracturing process may be used. The basic governingequations may include, for example, equations governing fluid flow inthe fracture network, the equation governing the fracture deformation,and the fracture propagation/interaction criterion.

Continuity equation assumes that fluid flow propagates along a fracturenetwork with the following mass conservation:

$\begin{matrix}{{\frac{\partial q}{\partial s} + \frac{\partial\left( {H_{fl}\overset{\_}{w}} \right)}{\partial t} + q_{L}} = 0} & (1)\end{matrix}$

where q is the local flow rate inside the hydraulic fracture along thelength, w is an average width or opening at the cross-section of thefracture at position s=s(x,y), H_(fl) is the height of the fluid in thefracture, and q_(L) is the leak-off volume rate through the wall of thehydraulic fracture into the matrix per unit height (velocity at whichfracturing fluid infiltrates into surrounding permeable medium) which isexpressed through Carter's leak-off model. The fracture tips propagateas a sharp front, and the length of the hydraulic fracture at any giventime t is defined as l(t).

The properties of driving fluid may be defined by power-law exponent n′(fluid behavior index) and consistency index K′. The fluid flow could belaminar, turbulent or Darcy flow through a proppant pack, and may bedescribed correspondingly by different laws. For the general case of 1Dlaminar flow of power-law fluid in any given fracture branch, thePoiseuille law (see, e.g., Nolte, 1991) may be used:

$\begin{matrix}{\frac{\partial p}{\partial s} = {{- \alpha_{0}}\frac{1}{{\overset{\_}{w}}^{{2\; n^{\prime}} + 1}}\frac{q}{H_{fl}}{\frac{q}{H_{fl}}}^{n^{\prime} - 1}}} & (2) \\{where} & \; \\{{{\alpha_{0} = {\frac{2\; K^{\prime}}{{\varphi \left( n^{\prime} \right)}^{n^{\prime}}} \cdot \left( \frac{{4\; n^{\prime}} + 2}{n^{\prime}} \right)^{n^{\prime}}}};}{{\varphi \left( n^{\prime} \right)} = {\frac{1}{H_{fl}}{\int_{H_{fl}}{\left( \frac{w(z)}{\overset{\_}{w}} \right)^{\frac{{2\; n^{\prime}} + 1}{n^{\prime}}}\ {z}}}}}} & (3)\end{matrix}$

Here w(z) represents fracture width as a function of depth at currentposition s, α is coefficient, n′ is power law exponent (fluidconsistency index), φ is shape function, and dz is the integrationincrement along the height of the fracture in the formula.

Fracture width may be related to fluid pressure through the elasticityequation. The elastic properties of the rock (which may be considered asmostly homogeneous, isotropic, linear elastic material) may be definedby Young's modulus E and Poisson's ratio v. For a vertical fracture in alayered medium with variable minimum horizontal stress σ_(h)(x, y, z)and fluid pressure p, the width profile (w) can be determined from ananalytical solution given as:

w(x,y,z)=w(p(x,y),H,z)  (4)

where W is the fracture width at a point with spatial coordinates x, y,z (coordinates of the center of fracture element); p(x,y) is the fluidpressure, H is the fracture element height, and z is the verticalcoordinate along fracture element at point (x,y).

Because the height of the fractures may vary, the set of governingequations may also include the height growth calculation as described,for example, in Kresse 2011.

In addition to equations described above, the global volume balancecondition may be satisfied:

$\begin{matrix}{{\int_{0}^{t}{{Q(t)}\ {t}}} = {{\int_{0}^{L{(t)}}{{H\left( {s,t} \right)}{\overset{\_}{w}\left( {s,t} \right)}\ {s}}} + {\int_{H_{L}}^{\;}{\int_{0}^{t}{\int_{0}^{L{(t)}}{2\; g_{L}\ {s}\ {t}\ {h_{l}}}}}}}} & (5)\end{matrix}$

where g_(L) is fluid leakoff velocity, Q(t) is time dependent injectionrate, H(s,t) height of the fracture at spacial point s(x,y) and at thetime t, ds is length increment for integration along fracture length,d_(t) is time increment, dh_(I) is increment of leakoff height, H_(L) isleakoff height, an s₀ is a spurt loss coefficient. Equation (5) providesthat the total volume of fluid pumped during time t is equal to thevolume of fluid in the fracture network and the volume leaked from thefracture up to time t. Here L(t) represents the total length of the HFNat the time t and S₀ is the spurt loss coefficient. The boundaryconditions may require the flow rate, net pressure and fracture width tobe zero at all fracture tips.

The system of Eq. 1-5, together with initial and boundary conditions,may be used to represent a set of governing equations. Combining theseequations and discretizing the fracture network into small elements maylead to a nonlinear system of equations in terms of fluid pressure p ineach element, simplified as f(p)=0, which may be solved by using adamped Newton-Raphson method.

Fracture interaction may be taken into account to model hydraulicfracture propagation in naturally fractured reservoirs. This includes,for example, the interaction between hydraulic fractures and naturalfractures, as well as interaction between hydraulic fractures. For theinteraction between hydraulic and natural fractures a semi-analyticalcrossing criterion may be implemented in the UFM using, for example, theapproach described in Gu and Weng 2010, and Gu et al. 2011.

Modeling of Stress Shadow

For parallel fractures, the stress shadow can be represented by thesuperposition of stresses from neighboring fractures. FIG. 2 is aschematic depiction of a 2D fracture 200 about a coordinate systemhaving an x-axis and a y-axis. Various points along the 2D fractures,such as a first end at h/2, a second end at −h/2 and a midpoint areextended to an observation point (x,y). Each line L extends at anglesθ₁, θ₂ from the points along the 2D fracture to the observation point.

The stress field around a 2D fracture with internal pressure p can becalculated using, for example, the techniques as described in Warpinskiand Teufel. The stress that affects fracture width is σ_(x), and can becalculated from:

$\begin{matrix}{{\sigma_{x} = {p\left\lbrack {1 - {\frac{\overset{\_}{L}}{\sqrt{{\overset{\_}{L}}_{1}{\overset{\_}{L}}_{2}}}{\cos \left( {\theta - \frac{\theta_{1} + \theta_{2}}{2}} \right)}} - {\frac{\overset{\_}{L}}{\left( {{\overset{\_}{L}}_{1}{\overset{\_}{L}}_{2}} \right)^{\frac{3}{2}}}\sin \; \theta \; {\sin \left( {\frac{3}{2}\left( {\theta_{1} + \theta_{2}} \right)} \right)}}} \right\rbrack}}\mspace{79mu} {where}} & (6) \\{\mspace{79mu} {{\theta = {\arctan \left( {- \frac{\overset{\_}{x}}{\overset{\_}{y}}} \right)}}\mspace{79mu} {\theta_{1} = {\arctan \left( {- \frac{\overset{\_}{x}}{\overset{\_}{1 + \overset{\_}{y}}}} \right)}}\mspace{79mu} {\theta_{2} = {\arctan \left( \frac{\overset{\_}{x}}{1 - \overset{\_}{y}} \right)}}}} & (7)\end{matrix}$

and where σ_(x) is stress in the x direction, p is internal pressure,and x, y, L, L ₁, L ₂ are the coordinates and distances in FIG. 2normalized by the fracture half-height h/2. Since σ_(x) varies in they-direction as well as in the x-direction, an averaged stress over thefracture height may be used in the stress shadow calculation.

The analytical equation given above can be used to compute the averageeffective stress of one fracture on an adjacent parallel fracture andcan be included in the effective closure stress on that fracture.

For more complex fracture networks, the fractures may orient indifferent directions and intersect each other. FIG. 3 shows a complexfracture network 300 depicting stress shadow effects. The fracturenetwork 300 includes hydraulic fractures 303 extending from a wellbore304 and interacting with other fractures 305 in the fracture network300.

A more general approach may be used to compute the effective stress onany given fracture branch from the rest of the fracture network. In UFM,the mechanical interactions between fractures may be modeled based on anenhanced 2D Displacement Discontinuity Method (DDM) (Olson 2004) forcomputing the induced stresses (see, e.g., FIG. 3).

In a 2D, plane-strain, displacement discontinuity solution, (see, e.g.,Crouch, S. L. and Starfield, A. M., Boundary Element Methods in SolidMechanics, George Allen & Unwin Ltd, London. Fisher, M. K.(1983)(hereafter Crouch and Starfield 1983), the entire contents ofwhich are hereby incorporated by reference) may be used to describe thenormal and shear stresses (σ_(n) and σ_(s)) acting on one fractureelement induced by the opening and shearing displacement discontinuities(D_(n) and D_(s)) from all fracture elements. To account for the 3Deffect due to finite fracture height, Olson 2004 may be used to providea 3D correction factor to the influence coefficients in combination withthe modified elasticity equations of 2D DDM as follows:

$\begin{matrix}{{\sigma_{n}^{i} = {{\sum\limits_{j = 1}^{N}{A^{ij}C_{n\; s}^{ij}D_{s}^{j}}} + {\sum\limits_{j = 1}^{N}{A^{ij}C_{nn}^{ij}D_{n}^{j}}}}}{\sigma_{s}^{i} = {{\sum\limits_{j = 1}^{N}{A^{ij}C_{ss}^{\; {ij}}D_{s}^{j}}} + {\sum\limits_{j = 1}^{N}{A^{ij}C_{sn}^{\; {ij}}D_{n}^{j}}}}}} & (8)\end{matrix}$

where A is a matrix of influence coefficients described in eq. (9), N isa total number of elements in the network whose interaction isconsidered, i is the element considered, and j=1, N are other elementsin the network whose influence on the stresses on element i arecalculated; and where C^(ij) are the 2D, plane-strain elastic influencecoefficients. These expressions can be found in Crouch and Starfield1983.

Elem i and j of FIG. 3 schematically depict the variables i and j inequation (8). Discontinuities D_(s) and D_(n) applied to Elem j are alsodepicted in FIG. 3. Dn may be the same as the fracture width, and theshear stress s may be 0 as depicted. Displacement discontinuity fromElem j creates a stress on Elem i as depicted by σ_(s) and σ_(n).

The 3D correction factor suggested by Olson 2004 may be presented asfollows:

$\begin{matrix}{A^{ij} = {1 - \frac{d_{ij}^{\beta}}{\left\lbrack {d_{ij}^{2} + \left( {h/\alpha} \right)^{2}} \right\rbrack^{\beta/2}}}} & (9)\end{matrix}$

where h is the fracture height, d_(ij) is the distance between elementsi and j, α and β are fitting parameters. Eq. 9 shows that the 3Dcorrection factor may lead to decaying of interaction between any twofracture elements when the distance increases.

In the UFM model, at each time step, the additional induced stresses dueto the stress shadow effects may be computed. It may be assumed that atany time, fracture width equals the normal displacement discontinuities(D_(n)) and shear stress at the fracture surface is zero, i.e., D_(n)^(j)=w_(j), σ_(s) ^(i)=0. Substituting these two conditions into Eq. 8,the shear displacement discontinuities (D_(s)) and normal stress inducedon each fracture element (σ_(n)) may be found.

The effects of the stress shadow induced stresses on the fracturenetwork propagation pattern may be described in two folds. First, duringpressure and width iteration, the original in-situ stresses at eachfracture element may be modified by adding the additional normal stressdue to the stress shadow effect. This may directly affect the fracturepressure and width distribution which may result in a change on thefracture growth. Second, by including the stress shadow induced stresses(normal and shear stresses), the local stress fields ahead of thepropagating tips may also be altered which may cause the local principalstress direction to deviate from the original in-situ stress direction.This altered local principal stress direction may result in the fractureturning from its original propagation plane and may further affect thefracture network propagation pattern.

Validation of Stress Shadow Model

Validation of the UFM model for the cases of bi-wing fractures may beperformed using, for example, Weng 2011 or Kresse 2011. Validation mayalso be performed using the stress shadow modeling approach. By way ofexample, the results may be compared using 2D DDM to Flac 3D as providedin Itasca Consulting Group Inc., 2002, FLAC3D (Fast Lagrangian Analysisof Continua in 3 Dimensions), Version 2.1, Minneapolis: ICG (2002)(hereafter “Itasca, 2002”).

Comparison of Enchansed 2D DDM to Flac3D

The 3D correction factors suggested by Olson 2004 contain two empiricalconstants, α and β. The values of α and β may be calibrated by comparingstresses obtained from numerical solutions (enhanced 2D DDM) to theanalytical solution for a plane-strain fracture with infinite length andfinite height. The model may further be validated by comparing the 2DDDM results to a full three dimensional numerical solutions, utilizing,for example, FLAC3D, for two parallel straight fractures with finitelengths and heights.

The validation problem is shown in FIG. 4. FIG. 4 a schematic diagram400 comparing enhanced 2D DDM to Flac3D for two parallel straightfractures. As shown in FIG. 400, two parallel fractures 407.1, 407.2 aresubject to stresses σ_(x), σ_(y) along an x, y coordinate axis. Thefractures have length 2L_(xf), and pressure of the fracture p₁, p₂,respectively. The fractures are a distance s apart.

The fracture in Flac3D may be simulated as two surfaces at the samelocation but with un-attached grid points. Constant internal fluidpressure may be applied as the normal stress on the grids. Fractures mayalso be subject to remote stresses, σ_(x) and σ_(y). Two fractures mayhave the same length and height with the ratio ofheight/half-length=0.3.

Stresses along x-axis (y=0) and y-axis (x=0) may be compared. Twoclosely spaced fractures (s/h=0.5) may be simulated as shown in thecomparison of FIGS. 5.1-5.3. These figures provide a comparison ofextended 2D DDM to Flac3D: Stresses along x-axis (y=0) and y-axis (x=0).

These figures include graphs 500.1, 500.2, 500.3, respectively,illustrating 2D DDM and Flac3D of extended fractures for σy along they-axis, σx along the y-axis, and σy along the x-axis, respectively. FIG.5.1 plots σy/p (y-axis) versus normalized distance from fracture(x-axis) using 2D DDM and Flac3D. FIG. 5.2 plots σx/p (y-axis) versusnormalized distance from fracture (x-axis) using 2D DDM and Flac3D. FIG.5.3 plots σy/p (y-axis) versus normalized distance from fracture(x-axis) using 2D DDM and Flac3D. The location L_(f) of the fracture tipis depicted along line x/h.

As shown in FIGS. 5.1-5.3, the stresses simulated from enhanced 2D DDMapproach with 3D correction factor match pretty well to those from thefull 3D simulator results, which indicates that the correction factorallows capture the 3D effect from the fracture height on the stressfield.

Comparison to CSIRO Model

The UFM model that incorporates the enchanced 2DDM approach may bevalidated against full 2D DDM simulator by CSIRO (see, e.g., Zhang, X.,Jeffrey, R. G., and Thiercelin, M. 2007. Deflection and Propagation ofFluid-Driven Fractures at Frictional Bedding Interfaces: A NumericalInvestigation. Journal of Structural Geology, 29: 396-410, (hereafter“Zhang 2007”) the entire contents of which is hereby incorporated byreference in its entirety). This approach may be used, for example, inthe limiting case of very large fracture height where 2D DDM approachesdo not consider 3D effects of the fractures height.

The comparison of influence of two closely propagating fractures on eachother's propagation paths may be employed. The propagation of twohydraulic fractures initiated parallel to each other (propagating alonglocal max stress direction) may be simulated for configurations, suchas: 1) initiation points on top of each other and offset from each otherfor isotropic, and 2) anisotropic far field stresses. The fracturepropagation path and pressure inside of each fracture may be comparedfor UFM and CSIRO code for the input data given in Table 1.

TABLE 1 Input data for validation against CSIRO model Injection rate0.106 m3/s 40 bbl/min Stress anisotropy 0.9 MPa 130 psi Young's modulus3 × 10{circumflex over ( )}10 Pa 4.35e+6 psi Poisson's ratio 0.35 0.35Fluid viscosity 0.001 pa-s 1 cp Fluid Specific 1.0  1.0  Gravity Minhorizontal stress 46.7 MPa 6773 psi Max horizontal 47.6 MPa 6903 psistress Fracture toughness 1 MPa-m^(0.5) 1000 psi/in^(0.5) Fractureheight 120 m 394 ft

When two fractures are initiated parallel to each other with initiationpoints separated by dx=0, dy=33 ft (10.1 m) (max horizontal stress fieldis oriented in x-direction), they may turn away from each other due tothe stress shadow effect.

The propagation paths for isotropic and anisotropic stress fields areshown in FIGS. 6.1 and 6.2. These figures are graphs 600.1, 600.2depicting propagation paths for two initially parallel fractures 609.1,609.2 in isotropic and anisotropic stress fields, respectively. Thefractures 609.1 and 609.2 are initially parallel near the injectionpoints 615.1, 615.2, but diverge as they extend away therefrom.Comparing with isotropic case, the curvatures of the fractures in thecase of stress anisotropy are depicted as being smaller. This may be dueto the competition between the stress shadow effect which tends to turnfractures away from each other, and far-field stresses which pushesfractures to propagate in the direction of maximum horizontal stress(x-direction). The influence of far-field stress becomes dominant as thedistance between the fractures increases, in which case the fracturesmay tend to propagate parallel to maximum horizontal stress direction.

FIGS. 7.1 and 7.2 depict graphs 700.1, 7002 showing a pair of fracturesinitiated from two different injection points 711.1, 711.2,respectively. These figures show a comparison for the case whenfractures are initiated from points separated by a distance dx=dy=(10.1m) for an isotropic and anisotropic stress field, respectively. In thesefigures, the fractures 709.1, 709.2 tend to propagate towards eachother. Examples of similar type of behavior have been observed in labexperiments (see, e.g., Zhang 2007).

As indicated above, the enchanced 2D DDM approach implemented in UFMmodel may be able to capture the 3D effects of finite fracture height onfracture interaction and propagation pattern, while beingcomputationally efficient. A good estimation of the stress field for anetwork of vertical hydraulic fractures and fracture propagationdirection (pattern) may be provided.

Example Cases Case #1 Parallel Fractures in Horizontal Wells

FIG. 8 is a schematic plot 800 of parallel transverse fractures 811.1,811.2, 811.3 propagating simultaneously from multiple perforationclusters 815.1, 815.2, 815.3, respectively, about a horizontal wellbore804. Each of the fractures 811.1, 811.2, 811.3 provides a different flowrate q₁, q₂, q₃ that is part of the total flow q_(t) at a pressure p₀.

When the formation condition and the perforations are the same for allthe fractures, the fractures may have about the same dimensions if thefriction pressure in the wellbore between the perforation clusters isproportionally small. This may be assumed where the fractures areseparated far enough and the stress shadow effects are negligible. Whenthe spacing between the fractures is within the region of stress shadowinfluence, the fractures may be affected not only in width, but also inother fracture dimension. To illustrate this, a simple example of fiveparallel fractures may be considered.

In this example, the fractures are assumed to have a constant height of100 ft (30.5 m). The spacing between the fractures is 65 ft (19.8 m).Other input parameters are given in Table 2.

TABLE 2 Input parameters for Case #1 Young's modulus 6.6 × 10⁶ psi =4.55e+10 Pa Poisson's ratio 0.35 Rate 12.2 bbl/min = 0.032 m3/sViscosity 300 cp = 0.3 Pa-s Height 100 ft = 30.5 m Leakoff coefficient3.9 × 10⁻² m/s^(1/2) Stress anisotropy 200 psi = 1.4 Mpa Fracturespacing 65 ft = 19.8 m No. of perfs per frac 100For this simple case, a conventional Perkins-Kern-Nordgren (PKN) model(see, e.g., Mack, M. G. and Warpinski, N. R., Mechanics of HydraulicFracturing. Chapter 6, Reservoir Stimulation, 3rd Ed., eds. Economides,M. J. and Nolte, K. G. John Wiley & Sons (2000)) for multiple fracturesmay be modified by incorporating the stress shadow calculation as givenfrom Eq. 6. The increase in closure stress may be approximated byaveraging the computed stress from Eq. 6 over the entire fracture. Notethat this simplistic PKN model may not simulate the fracture turning dueto the stress shadow effect. The results from this simple model may becompared to the results from the UFM model that incorporatespoint-by-point stress shadow calculation along the entire fracture pathsas well as fracture turning.

FIG. 9 shows the simulation results of fracture lengths of the fivefractures, computed from both models. FIG. 9 is a graph 900 depictinglength (y-axis) versus time (t) of five parallel fractures duringinjection. Lines 917.1-917.5 are generated from the UFM model. Lines919.1-919.5 are generated from the simplistic PKN model.

The fracture geometry and width contour from the UFM model for the fivefractures of FIG. 9 are shown in FIG. 10. FIG. 10 is a schematic diagram1000 depicting fractures 1021.1-1021.5 about a wellbore 1004.

Fracture 1021.3 is the middle one of the five fractures, and fractures1021.1 and 1021.5 are the outmost ones. Since fractures 1021.2, 1021.3,and 1021.4 have smaller width than that of the outer ones due to thestress shadow effect, they may have larger flow resistance, receive lessflow rate, and have shorter length. Therefore, the stress shadow effectsmay not only be fracture width but also fracture length under dynamicconditions.

The effect of stress shadow on fracture geometry may be influenced bymany parameters. To illustrate the effect of some of these parameters,the computed fracture lengths for the cases with varying fracturespacing, perforation friction, and stress anisotropy are shown in Table3.

FIGS. 11.1 and 11.2 shows the fracture geometry predicted by the UFM forthe case of large perforation friction and the case of large fracturespacing (e.g., about 120 ft (36.6 m)). FIGS. 11.1 and 11.2 are schematicdiagrams 1100.1 and 1100.2 depicting five fractures 1123.1-1123.5 abouta wellbore 1104. When the perforation friction is large, a largediversion force that uniformly distributes the flow rate into allperforation clusters may be provided. Consequently, the stress shadowmay be overcome and the resulting fracture lengths may becomeapproximately equal as shown in FIG. 11.1. When fracture spacing islarge, the effect of the stress shadow may dissipate, and fractures mayhave approximately the same dimensions as shown in FIG. 11.2.

TABLE 3 Influence of various parameters on fracture geometry Base 120 ftspacing No. of Anisotropy = 50 psi Frac case (36.6 m) perfs = 2 (345000Pa) 1 133 113 105 111 2 93 104 104 95 3 83 96 104 99 4 93 104 100 95 5123 113 109 102

Case #2 Complex Fractures

In an example of FIG. 12, the UFM model may be used to simulate a4-stage hydraulic fracture treatment in a horizontal well in a shaleformation. See, e.g., Cipolla, C., Weng, X., Mack, M., Ganguly, U.,Kresse, O., Gu, H., Cohen, C. and Wu, R., Integrating MicroseismicMapping and Complex Fracture Modeling to Characterize FractureComplexity. Paper SPE 140185 presented at the SPE Hydraulic FracturingConference and Exhibition, Woodlands, Tex., USA, Jan. 24-26, 2011,(hereinafter “Cipolla 2011”) the entire contents of which are herebyincorporated by reference in their entirety. The well may be cased andcemented, and each stage pumped through three or four perforationclusters. Each of the four stages may consist of approximately 25,000bbls (4000 m³) of fluid and 440,000 lbs (2e+6 kg) of proppant. Extensivedata may be available on the well, including advanced sonic logs thatprovide an estimate of minimum and maximum horizontal stress.Microseismic mapping data may be available for all stages. See, e.g,Daniels, J., Waters, G., LeCalvez, J., Lassek, J., and Bentley, D.,Contacting More of the Barnett Shale Through an Integration of Real-TimeMicroseismic Monitoring, Petrophysics, and Hydraulic Fracture Design.Paper SPE 110562 presented at the 2007 SPE Annual Technical Conferenceand Exhibition, Anaheim, Calif., USA, Oct. 12-14, 2007. This example isshown in FIG. 12. FIG. 12 is a graph depicting microseismic mapping ofmicroseismic events 1223 at various stages about a wellbore 1204.

The stress anisotropy from the advanced sonic log, indicates a higherstress anisotropy in the toe section of the well compared to the heel.An advanced 3D seismic interpretation may indicate that the dominantnatural fracture trend changes from NE-SW in the toe section to NW-SE inheel portion of the lateral. See, e.g., Rich, J. P. and Ammerman, M.,Unconventional Geophysics for Unconventional Plays. Paper SPE 131779presented at the Unconventional Gas Conference, Pittsburgh, Pa., USA,Feb. 23-25, 2010, the entire contents of which is hereby incorporated byreference herein in its entirety.

Simulation results may be based on the UFM model without incorporatingthe full stress shadow calculation (see, e.g., Cipolla 2011), includingshear stress and fracture turning (see, e.g., Weng 2011). The simulationmay be updated with the full stress model as provided herein. FIGS.13.1-13.4 show a plan view of a simulated fracture network 1306 about awellbore 1304 for all four stages, respectively, and their comparison tothe microseismic measurements 1323.1-1323.4, respectively.

From simulation results in FIGS. 13.1-13.4, it can be seen that forStages 1 and 2, the closely spaced fractures did not divergesignificantly. This may be because of the high stress anisotropy in thetoe section of the wellbore. For Stage 3 and 4, where stress anisotropyis lower, more fracture divergence can be seen as a result of the stressshadow effect.

Case #3 Multi-Stage Example

Case #3 is an example showing how stress shadow from previous stages caninfluence the propagation pattern of hydraulic fracture networks fornext treatment stages, resulting in changing of total picture ofgenerated hydraulic fracture network for the four stage treatment case.

This case includes four hydraulic fracture treatment stages. The well iscased and cemented. Stages 1 and 2 are pumped through three perforatedclusters, and Stages 3 and 4 are pumped through four perforatedclusters. The rock fabric is isotropic. The input parameters are listedin Table 4 below. The top view of total hydraulic fracture networkwithout and with accounting for stress shadow from previous stages isshown in FIGS. 13.1-13.4.

TABLE 4 Input parameters for Case #3 Young's modulus 4.5 × 10⁶ psi =3.1e+10 Pa Poisson's ratio 0.35 Rate 30.9 bpm = 0.082 m3/s Viscosity 0.5cp = 0.0005 pa-s Height 330 ft = 101 m Pumping time 70 min

FIGS. 14.1-14.4 are schematic diagrams 1400.1-1400-4 depicting afracture network 1429 at various stages during a fracture operation.FIG. 14.1 shows a discrete fracture network (DFN) 1429 before treatment.FIG. 14.2 depicts a simulated DFN 1429 after a first treatment stage.The DFN 1429 has propagated hydraulic fractures (HFN) 1431 extendingtherefrom due to the first treatment stage. FIG. 14.3 shows the DFNdepicting a simulated HFN 1431.1-1431.4 propagated during four stages,respectively, but without accounting for previous stage effects. FIG.14.4 shows the DFN depicting HFN 1431.1, 1431.2′-1431.4′ propagatedduring four stages, but with accounting for the fractures, stressshadows and HFN from previous stages.

When stages are generated separately, they may not see each other asindicated in FIG. 14.3. When stress shadow and HFN from previous stagesare taken into account as in FIG. 14.4 the propagation pattern maychange. The hydraulic fractures 1431.1 generated for the first stage isthe same for both case scenarios as shown in FIGS. 14.3 and 14.4. Thesecond stage 1431.2 propagation pattern may be influenced by the firststage through stress shadow, as well as through new DFN (including HFN1431.1 from Stage 1), resulting in the changing of propagation patternsto HFN 1431.2′. The HFN 1431.1′ may start to follow HFN 1431.1 createdat stage 1 while intercounting it. The third stage 1431.3 may follow ahydraulic fracture created during second stage treatment 1431.2,1431.2′, and may not propagate too far due to stress shadow effect fromStage 2 as indicated by 1431.3 versus 1431.3′. Stage 4 (1431.4) may tendto turn away from stage three when it could, but may follow HFN 1431.3′from previous stages when encounters it and be depicted as HFN 1431.4′in FIG. 14.4.

A method for computing the stress shadow in a complex hydraulic fracturenetwork is presented. The method may involve an enhanced 2D or 3DDisplacement Discontinuity Method with correction for finite fractureheight. The method may be used to approximate the interaction betweendifferent fracture branches in a complex fracture network for thefundamentally 3D fracture problem. This stress shadow calculation may beincorporated in the UFM, a complex fracture network model. The resultsfor simple cases of two fractures show the fractures can either attractor expel each other depending on their initial relative positions, andcompare favorably with an independent 2D non-planar hydraulic fracturemodel.

Simulations of multiple parallel fractures from a horizontal well may beused to confirm the behavior of the two outmost fractures that may bemore dominant, while the inner fractures have reduced fracture lengthand width due to the stress shadow effect. This behavior may also dependon other parameters, such as perforation friction and fracture spacing.When fracture spacing is greater than fracture height, the stress shadoweffect may diminish and there may be insignificant differences among themultiple fractures. When perforation friction is large, sufficientdiversion to distribute the flow equally among the perforation clustersmay be provided, and the fracture dimensions may become approximatelyequal despite the stress shadow effect.

When complex fractures are created, if the formation has a small stressanisotropy, fracture interaction can lead to dramatic divergence of thefractures where they tend to repel each other. On the other hand, forlarge stress anisotropy, there may be limited fracture divergence wherethe stress anisotropy offsets the effect of fracture turning due to thestress shadow, and the fracture may be forced to go in the direction ofmaximum stress. Regardless of the amount of fracture divergence, thestress shadowing may have an effect on fracture width, which may affectthe injection rate distribution into multiple perforation clusters, andoverall fracture network footprint and proppant placement.

FIG. 15 is a flow chart depicting a method 1500 of performing a fractureoperation at a wellsite, such as the wellsite 100 of FIG. 1.1. Thewellsite is positioned about a subterranean formation having a wellboretherethrough and a fracture network therein. The fracture network hasnatural fractures as shown in FIGS. 1.1 and 1.2. The method (1500) mayinvolve (1580) performing a stimulation operation by stimulating thewellsite by injection of an injection fluid with proppant into thefracture network to form a hydraulic fracture network. In some cases,the stimulation may be performed at the wellsite or by simulation.

The method involves (1582) obtaining wellsite data and a mechanicalearth model of the subterranean formation. The wellsite data may includeany data about the wellsite that may be useful to the simulation, suchas natural fracture parameters of the natural fractures, images of thefracture network, etc. The natural fracture parameters may include, forexample, density orientation, distribution, and mechanical properties(e.g., coefficients of friction, cohesion, fracture toughness, etc.) Thefracture parameters may be obtained from direct observations of boreholeimaging logs, estimated from 3D seismic, ant tracking, sonic waveanisotropy, geological layer curvature, microseismic events or images,etc. Examples of techniques for obtaining fracture parameters areprovided in PCT/US2012/48871 and US2008/0183451, the entire contents ofwhich are hereby incorporated by reference herein in their entirety.

Images may be obtained by, for example, observing borehole imaging logs,estimating fracture dimensions from wellbore measurements, obtainingmicroseismic images, and/or the like. The fracture dimensions may beestimated by evaluating seismic measurements, ant tracking, sonicmeasurements, geological measurements, and/or the like. Other wellsitedata may also be generated from various sources, such as wellsitemeasurements, historical data, assumptions, etc. Such data may involve,for example, completion, geological structure, petrophysical,geomechanical, log measurement and other forms of data. The mechanicalearth model may be obtained using conventional techniques.

The method (1500) also involves (1584) generating a hydraulic fracturegrowth pattern over time, such as during the stimulation operation.FIGS. 16.1-16.4 depict an example of (1584) generating a hydraulicfracture growth pattern. As shown in FIG. 16.1, in its initial state, afracture network 1606.1 with natural fractures 1623 is positioned abouta subterranean formation 1602 with a wellbore 1604 therethrough. Asproppant is injected into the subterranean formation 1602 from thewellbore 1604, pressure from the proppant creates hydraulic fractures1691 about the wellbore 1604. The hydraulic fractures 1691 extend intothe subterranean formation along L₁ and L₂ (FIG. 16.2), and encounterother fractures in the fracture network 1606.1 over time as indicated inFIGS. 16.2-16.3. The points of contact with the other fractures areintersections 1625.

The generating (1584) may involve (1586) extending hydraulic fracturesfrom the wellbore and into the fracture network of the subterraneanformation to form a hydraulic fracture network including the naturalfractures and the hydraulic fractures as shown in FIG. 16.2. Thefracture growth pattern is based on the natural fracture parameters anda minimum stress and a maximum stress on the subterranean formation. Thegenerating may also involve (1588) determining hydraulic fractureparameters (e.g., pressure p, width w, flow rate q, etc.) of thehydraulic fractures, (1590) determining transport parameters for theproppant passing through the hydraulic fracture network, and (1592)determining fracture dimensions (e.g., height) of the hydraulicfractures from, for example, the determined hydraulic fractureparameters, the determined transport parameters and the mechanical earthmodel. The hydraulic fracture parameters may be determined after theextending. The determining (1592) may also be performed by from theproppant transport parameters, wellsite parameters and other items.

The generating (1584) may involve modeling rock properties based on amechanical earth model as described, for example, in Koutsabeloulis andZhang, 3D Reservoir Geomechanics Modeling in Oil/Gas Field Production,SPE Paper 126095, 2009 SPE Saudi Arabia Section Technical Symposium andExhibition held in Al Khobar, Saudi Arabia, 9-11 May, 2009. Thegenerating may also involve modeling the fracture operation by using thewellsite data, fracture parameters and/or images as inputs modelingsoftware, such as UFM, to generate successive images of inducedhydraulic fractures in the fracture network.

The method (1500) also involves (1594) performing stress shadowing onthe hydraulic fractures to determine stress interference between thehydraulic fractures (or with other fractures), and (1598) repeating thegenerating (1584) based on the stress shadowing and/or the determinedstress interference between the hydraulic fractures. The repeating maybe performed to account for fracture interference that may affectfracture growth. Stress shadowing may involve performing, for example, a2D or 3D DDM for each of the hydraulic fractures and updating thefracture growth pattern over time. The fracture growth pattern maypropagate normal to a local principal stress direction according tostress shadowing. The fracture growth pattern may involve influences ofthe natural and hydraulic fractures over the fracture network (see FIG.16.3).

Stress shadowing may be performed for multiple wellbores of thewellsite. The stress shadowing from the various wellbores may becombined to determine the interaction of fractures as determined fromeach of the wellbores. The generating may be repeated for each of thestress shadowings performed for one or more of the multiple wellbores.The generating may also be repeated for stress shadowing performed wherestimulation is provided from multiple wellbores. Multiple simulationsmay also be performed on the same wellbore with various combinations ofdata, and compared as desired. Historical or other data may also beinput into the generating to provide multiple sources of information forconsideration in the ultimate results.

The method also involves (1596) determining crossing behavior betweenthe hydraulic fractures and an encountered fracture if the hydraulicfracture encounters another fracture, and (1598) repeating thegenerating (1584) based on the crossing behavior if the hydraulicfracture encounters a fracture (see, e.g., FIG. 16.3). Crossing behaviormay be determined using, for example, the techniques ofPCT/US2012/059774, the entire contents of which is hereby incorporatedherein in its entirety.

The determining crossing behavior may involve performing stressshadowing. Depending on downhole conditions, the fracture growth patternmay be unaltered or altered when the hydraulic fracture encounters thefracture. When a fracture pressure is greater than a stress acting onthe encountered fracture, the fracture growth pattern may propagatealong the encountered fracture. The fracture growth pattern may continuepropagation along the encountered fracture until the end of the naturalfracture is reached. The fracture growth pattern may change direction atthe end of the natural fracture, with the fracture growth patternextending in a direction normal to a minimum stress at the end of thenatural fracture as shown in FIG. 16.4. As shown in FIG. 16.4, thehydraulic fracture extends on a new path 1627 according to the localstresses σ₁ and σ₂.

Optionally, the method (1500) may also involve (1599) validating thefracture growth pattern. The validation may be performed by comparingthe resulting growth pattern with other data, such as microseismicimages as shown, for example, in FIGS. 7.1 and 7.2.

The method may be performed in any order and repeated as desired. Forexample, the steps of generating (1584)-(1599) may be repeated overtime, for example, by iteration as the fracture network changes. Thegenerating (1584) may be performed to update the iterated simulationperformed during the generating to account for the interaction andeffects of multiple fractures as the fracture network is stimulated overtime.

Although the present disclosure has been described with reference toexemplary embodiments and implementations thereof, the presentdisclosure is not to be limited by or to such exemplary embodimentsand/or implementations. Rather, the systems and methods of the presentdisclosure are susceptible to various modifications, variations and/orenhancements without departing from the spirit or scope of the presentdisclosure. Accordingly, the present disclosure expressly encompassesall such modifications, variations and enhancements within its scope.

It should be noted that in the development of any such actualembodiment, or numerous implementation, specific decisions must be madeto achieve the developer's specific goals, such as compliance withsystem related and business related constraints, which will vary fromone implementation to another. Moreover, it will be appreciated thatsuch a development effort might be complex and time consuming but wouldnevertheless be a routine undertaking for those of ordinary skill in theart having the benefit of this disclosure. In addition, the embodimentsused/disclosed herein can also include some components other than thosecited.

In the description, each numerical value should be read once as modifiedby the term “about” (unless already expressly so modified), and thenread again as not so modified unless otherwise indicated in context.Also, in the description, it should be understood that any range listedor described as being useful, suitable, or the like, is intended thatany and every value within the range, including the end points, is to beconsidered as having been stated. For example, “a range of from 1 to 10”is to be read as indicating each and every possible number along thecontinuum between about 1 and about 10. Thus, even if specific datapoints within the range, or even no data points within the range, areexplicitly identified or refer to only a few specific ones, it is to beunderstood that inventors appreciate and understand that any and alldata points within the range are to be considered to have beenspecified, and that inventors possessed knowledge of the entire rangeand all points within the range.

The statements made herein merely provide information related to thepresent disclosure and may not constitute prior art, and may describesome embodiments illustrating the invention. All references cited hereinare incorporated by reference into the current application in theirentirety.

Although only a few example embodiments have been described in detailabove, those skilled in the art will readily appreciate that manymodifications are possible in the example embodiments without materiallydeparting from the system and method for performing wellbore stimulationoperations. Accordingly, all such modifications are intended to beincluded within the scope of this disclosure as defined in the followingclaims. In the claims, means-plus-function clauses are intended to coverthe structures described herein as performing the recited function andnot only structural equivalents, but also equivalent structures. Thus,although a nail and a screw may not be structural equivalents in that anail employs a cylindrical surface to secure wooden parts together,whereas a screw employs a helical surface, in the environment offastening wooden parts, a nail and a screw may be equivalent structures.It is the express intention of the applicant not to invoke 35 U.S.C.§112, paragraph 6 for any limitations of any of the claims herein,except for those in which the claim expressly uses the words ‘means for’together with an associated function.

What is claimed is:
 1. A method of performing a fracture operation at awellsite, the wellsite positioned about a subterranean formation havinga wellbore therethrough and a fracture network therein, the fracturenetwork comprising natural fractures, the wellsite stimulated byinjection of an injection fluid with proppant into the fracture network,the method comprising: obtaining wellsite data comprising naturalfracture parameters of the natural fractures and obtaining a mechanicalearth model of the subterranean formation; generating a hydraulicfracture growth pattern for the fracture network over time, thegenerating comprising: extending hydraulic fractures from the wellboreand into the fracture network of the subterranean formation to form ahydraulic fracture network comprising the natural fractures and thehydraulic fractures; determining hydraulic fracture parameters of thehydraulic fractures after the extending; determining transportparameters for the proppant passing through the hydraulic fracturenetwork; and determining fracture dimensions of the hydraulic fracturesfrom the determined hydraulic fracture parameters, the determinedtransport parameters and the mechanical earth model; and performingstress shadowing on the hydraulic fractures to determine stressinterference between the hydraulic fractures; and repeating thegenerating based on the determined stress interference.
 2. The method ofclaim 1, further comprising if the hydraulic fractures encounter anotherfracture, determining crossing behavior at the encountered anotherfracture, and wherein the repeating comprises repeating the generatingbased on the determined stress interference and the crossing behavior.3. The method of claim 2, wherein the hydraulic fracture growth patternis unaltered by the crossing behavior.
 4. The method of claim 2, whereinthe fracture growth pattern is altered by the crossing behavior.
 5. Themethod of claim 2, wherein a fracture pressure of the hydraulic fracturenetwork is greater than a stress acting on the encountered fracture andwherein the fracture growth pattern propagates along the encounteredfracture.
 6. The method of claim 1, wherein the fracture growth patterncontinues to propagate along the encountered fracture until an end ofthe natural fracture is reached.
 7. The method of claim 1, wherein thefracture growth pattern changes direction at the end of the naturalfracture, the fracture growth pattern extending in a direction normal toa minimum stress at the end of the natural fracture.
 8. The method ofclaim 1, wherein the fracture growth pattern propagates normal to alocal principal stress according to the stress shadowing.
 9. The methodof claim 1, wherein the stress shadowing comprises performingdisplacement discontinuity for each of the hydraulic fractures.
 10. Themethod of claim 1, wherein the stress shadowing comprises performing thestress shadowing about multiple wellbores of a wellsite and repeatingthe generating using the stress shadowing performed on the multiplewellbores.
 11. The method of claim 1, wherein the stress shadowingcomprises performing the stress shadowing at multiple stimulation stagesin the wellbore.
 12. The method of claim 1, further comprisingvalidating the fracture growth pattern.
 13. The method of claim 12,wherein the validating comprises comparing the fracture growth patternwith at least one simulation of stimulation of the fracture network. 14.The method of claim 1, wherein the extending comprises extending thehydraulic fractures along the hydraulic fracture growth pattern based onthe natural fracture parameters and a minimum stress and a maximumstress on the subterranean formation.
 15. The method of claim 1, whereinthe determining fracture dimensions comprises one of evaluating seismicmeasurements, ant tracking, sonic measurements, geological measurementsand combinations thereof.
 16. The method of claim 1, wherein thewellsite data further comprises at least one of geological,petrophysical, geomechanical, log measurements, completion, historicaland combinations thereof.
 17. The method of claim 1, wherein the naturalfracture parameters are generated by one of observing borehole imaginglogs, estimating fracture dimensions from wellbore measurements,obtaining microseismic images, and combinations thereof.
 18. A method ofperforming a fracture operation at a wellsite, the wellsite positionedabout a subterranean formation having a wellbore therethrough and afracture network therein, the fracture network comprising naturalfractures, the wellsite stimulated by injection of an injection fluidwith proppant into the fracture network, the method comprising:obtaining wellsite data comprising natural fracture parameters of thenatural fractures and obtaining a mechanical earth model of thesubterranean formation; generating a hydraulic fracture growth patternfor the fracture network over time, the generating comprising: extendinghydraulic fractures from the wellbore and into the fracture network ofthe subterranean formation to form a hydraulic fracture networkcomprising the natural fractures and the hydraulic fractures;determining hydraulic fracture parameters of the hydraulic fracturesafter the extending; determining transport parameters for the proppantpassing through the hydraulic fracture network; and determining fracturedimensions of the hydraulic fractures from the determined hydraulicfracture parameters, the determined transport parameters and themechanical earth model; and performing stress shadowing on the hydraulicfractures to determine stress interference between the hydraulicfractures; if the hydraulic fracture encounters another fracture,determining crossing behavior between the hydraulic fractures and anencountered fracture based on the determined stress interference; andrepeating the generating based on the determined stress interference andthe crossing behavior.
 19. The method of claim 18, further comprisingvalidating the fracture growth pattern.
 20. A method of performing afracture operation at a wellsite, the wellsite positioned about asubterranean formation having a wellbore therethrough and a fracturenetwork therein, the fracture network comprising natural fractures, themethod comprising: stimulating the wellsite by injection of an injectionfluid with proppant into the fracture network; obtaining wellsite datacomprising natural fracture parameters of the natural fractures andobtaining a mechanical earth model of the subterranean formation;generating a hydraulic fracture growth pattern for the fracture networkover time, the generating comprising: extending hydraulic fractures fromthe wellbore and into the fracture network of the subterranean formationto form a hydraulic fracture network comprising the natural fracturesand the hydraulic fractures; determining hydraulic fracture parametersof the hydraulic fractures after the extending; determining transportparameters for the proppant passing through the hydraulic fracturenetwork; and determining fracture dimensions of the hydraulic fracturesfrom the determined hydraulic fracture parameters, the determinedtransport parameters and the mechanical earth model; and performingstress shadowing on the hydraulic fractures to determine stressinterference between the hydraulic fractures; and repeating thegenerating based on the determined stress interference.
 21. The methodof claim 20, further comprising validating the hydraulic fracture growthpattern.
 22. The method of claim 20, further comprising if the hydraulicfractures encounters another fracture, determining crossing behaviorbetween the hydraulic fractures and the encountered another fracture,and wherein the repeating comprises repeating the generating based onthe determined stress interference and the crossing behavior.